Simplify; express your answer in exponential form. Assume $a\neq 0, q\neq 0$. $\dfrac{{(a^{4}q^{5})^{-1}}}{{(a^{-1}q^{5})^{-5}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{4}q^{5})^{-1} = (a^{4})^{-1}(q^{5})^{-1}}$ On the left, we have ${a^{4}}$ to the exponent ${-1}$ . Now ${4 \times -1 = -4}$ , so ${(a^{4})^{-1} = a^{-4}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{4}q^{5})^{-1}}}{{(a^{-1}q^{5})^{-5}}} = \dfrac{{a^{-4}q^{-5}}}{{a^{5}q^{-25}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-4}q^{-5}}}{{a^{5}q^{-25}}} = \dfrac{{a^{-4}}}{{a^{5}}} \cdot \dfrac{{q^{-5}}}{{q^{-25}}} = a^{{-4} - {5}} \cdot q^{{-5} - {(-25)}} = a^{-9}q^{20}$